In our previous post, we looked at James McCaffrey’s code, “Gradient Descent Training Using C#” from MSDN magazine, and took a stab at rewriting the first part in F#, to clarify a bit the way the dataset was created. Today, we’ll dive in the second block, which implements the logistic regression using gradient descent. Again, we won’t discuss why the algorithm works – the article does a pretty good job at that – and focus instead purely on the F# / C# conversion part.

Let’s begin by taking a look at the core of the C# code, which lives in the LogisticClassifier class. I took the liberty to do some minor cleanup, and remove some parts which were un-necessary, so as to make it a bit easier to see what is going on:

public class LogisticClassifier { private int numFeatures; // number of x variables aka features private double[] weights; // b0 = constant private Random rnd; public LogisticClassifier(int numFeatures) { this.numFeatures = numFeatures; this.weights = new double[numFeatures + 1]; // [0] = b0 constant this.rnd = new Random(0); } public double[] Train(double[][] trainData, int maxEpochs, double alpha) { // alpha is the learning rate int epoch = 0; int[] sequence = new int[trainData.Length]; // random order for (int i = 0; i < sequence.Length; ++i) sequence[i] = i; while (epoch < maxEpochs) { ++epoch; if (epoch % 100 == 0 && epoch != maxEpochs) { double mse = Error(trainData, weights); Console.Write("epoch = " + epoch); Console.WriteLine(" error = " + mse.ToString("F4")); } Shuffle(sequence); // process data in random order // stochastic/online/incremental approach for (int ti = 0; ti < trainData.Length; ++ti) { int i = sequence[ti]; double computed = ComputeOutput(trainData[i], weights); int targetIndex = trainData[i].Length - 1; double target = trainData[i][targetIndex]; weights[0] += alpha * (target - computed) * 1; // the b0 weight has a dummy 1 input for (int j = 1; j < weights.Length; ++j) weights[j] += alpha * (target - computed) * trainData[i][j - 1]; } } // while return this.weights; // by ref is somewhat risky } // Train private void Shuffle(int[] sequence) { for (int i = 0; i < sequence.Length; ++i) { int r = rnd.Next(i, sequence.Length); int tmp = sequence[r]; sequence[r] = sequence[i]; sequence[i] = tmp; } } private double Error(double[][] trainData, double[] weights) { // mean squared error using supplied weights int yIndex = trainData[0].Length - 1; // y-value (0/1) is last column double sumSquaredError = 0.0; for (int i = 0; i < trainData.Length; ++i) // each data { double computed = ComputeOutput(trainData[i], weights); double desired = trainData[i][yIndex]; // ex: 0.0 or 1.0 sumSquaredError += (computed - desired) * (computed - desired); } return sumSquaredError / trainData.Length; } private double ComputeOutput(double[] dataItem, double[] weights) { double z = 0.0; z += weights[0]; // the b0 constant for (int i = 0; i < weights.Length - 1; ++i) // data might include Y z += (weights[i + 1] * dataItem[i]); // skip first weight return 1.0 / (1.0 + Math.Exp(-z)); } } // LogisticClassifier

Just from the length of it, you can tell that most of the action is taking place in the Train method, so let’s start there. What we have here is two nested loops. The outer one runs maxEpoch times, a user defined parameter. Inside that loop, we randomly shuffle the input dataset, and then loop over each training example, computing the predicted output of the logistic function for that example, comparing it to a target, the actual label of the example, which can be 0 or 1, and adjusting the weights so as to reduce the error. We also have a bit of logging going on, displaying the prediction error every hundred outer iteration. Once the two loops are over, we return the weights.

Two things strike me here. First, a ton of indexes are involved, and this tends to obfuscate what is going on; as a symptom, a few comments are needed, to clarify how the indexes work, and what piece of the data is organized. Then, there is a lot of mutation going on. It’s not necessarily a bad thing, but I tend to avoid it as much as possible, simply because it requires keeping more moving parts in my head when I try to follow the code, and also, as McCaffrey himself points out in a comment, because “by ref is somewhat risky”.

As a warm up, let’s begin with the error computation, which is displayed every 100 iterations. Rather than having to remember in what column the actual expected value is stored, let’s make our life easier, and use a type alias, Example, so that the features are neatly tucked in an array, and the value is clearly separated. We need to compute the average square difference between the expected value, and the output of the logistic function for each example. As it turns out, we have already implemented the logistic function in the first part in the code, so re-implementing it as in ComputeOutput seems like un-necessary work – we can get rid of that part entirely, and simply map every example to the square error, and compute the average, using pattern matching on the examples to separate clearly the features and the expected value:

type Example = float [] * float let Error (trainData:Example[], weights:float[]) = // mean squared error using supplied weights trainData |> Array.map (fun (features,value) -> let computed = logistic weights features let desired = value (computed - desired) * (computed - desired)) |> Array.average

Some of you might argue that this could be made tighter – I can think of at least two possibilities. First, using a Tuple might not be the most expressive approach; replacing it with a Record instead could improve readability. Then, we could also skip the map + average part, and directly ask F# to compute the average on the fly:

type Example = { Features:float[]; Label:float } let Error (trainData:Example[], weights:float[]) = trainData |> Array.averageBy (fun example -> let computed = logistic weights example.Features let desired = example.Label (computed - desired) * (computed - desired))

I will keep my original version the way it is, mostly because we created a dataset based on tuples last times.

We are now ready to hit the center piece of the algorithm. Just like we would probably try to extract a method in C#, we will start extracting some of the gnarly code that lies in the middle:

for (int ti = 0; ti < trainData.Length; ++ti) { int i = sequence[ti]; double computed = ComputeOutput(trainData[i], weights); int targetIndex = trainData[i].Length - 1; double target = trainData[i][targetIndex]; weights[0] += alpha * (target - computed) * 1; // the b0 weight has a dummy 1 input for (int j = 1; j < weights.Length; ++j) weights[j] += alpha * (target - computed) * trainData[i][j - 1]; }

Rather than modify the weights values, it seems safer to compute new weights. And because we opted last week to insert a column with ones for the constant feature, we won’t have to deal with the index misalignment, which requires separate handling for b0 and the rest. Instead, we can write an update operation that takes in an example and weights, and returns new weights:

let update (example:Example) (weights:float[]) = let features,target = example let computed = logistic weights features weights |> Array.mapi (fun i w -> w + alpha * (target - computed) * features.[i])

Array.mapi allows us to iterate over the weights, while maintaining the index we are currently at, which we use to grab the feature value at the corresponding index. Alternatively, you could go all verbose and zip the arrays together – or all fancy with a double-pipe and map2 to map the two arrays in one go. Your pick:

Array.zip weights features |> Array.map (fun (weight,feat) -> weight + alpha * (target - computed) * feat) (weights,features) ||> Array.map2 (fun weight feat -> weight + alpha * (target - computed) * feat)

We are now in a very good place; the only thing left to do is to plug that into the two loops. The inner loop is a perfect case for a fold (the Aggregate method in LINQ): given a starting value for weights, we want to go over every example in our training set, and, for each of them, run the update function to compute new weights. For the while loop, we’ll take a different approach, and use recursion: when the epoch reaches maxEpoch, you are done, return the weights, otherwise, keep shuffling the data and updating weights. Let’s put that all together:

let Train (trainData:Example[], numFeatures, maxEpochs, alpha, seed) = let rng = Random(seed) let epoch = 0 let update (example:Example) (weights:float[]) = let features,target = example let computed = logistic weights features weights |> Array.mapi (fun i w -> w + alpha * (target - computed) * features.[i]) let rec updateWeights (data:Example[]) epoch weights = if epoch % 100 = 0 then printfn "Epoch: %i, Error: %.2f" epoch (Error (data,weights)) if epoch = maxEpochs then weights else let data = shuffle rng data let weights = data |> Array.fold (fun w example -> update example w) weights updateWeights data (epoch + 1) weights // initialize the weights and start the recursive update let initialWeights = [| for _ in 1 .. numFeatures + 1 -> 0. |] updateWeights trainData 0 initialWeights

And that’s pretty much it. We replaced the whole class by a couple of functions, and all the indexes are gone. This is probably a matter of taste and comfort with functional concepts, but in my opinion, this is much easier to follow.

Before trying it out, to make sure it works, I’ll take a small liberty, and modify the Train function. As it stands right now, it returns the final weights, but really, we don’t care about the weights, what we want is a classifier, which is a function that, given an array, will predict a one or a zero. That’s easy enough, let’s return a function at the end instead of weights:

// initialize the weights and start the recursive update let initialWeights = [| for _ in 1 .. numFeatures + 1 -> 0. |] let finalWeights = updateWeights trainData 0 initialWeights let classifier (features:float[]) = if logistic finalWeights features > 0.5 then 1. else 0. classifier

We can now wrap it up, and see our code in action:

printfn "Begin Logistic Regression (binary) Classification demo" printfn "Goal is to demonstrate training using gradient descent" let numFeatures = 8 // synthetic data let numRows = 10000 let seed = 1 printfn "Generating %i artificial data items with %i features" numRows numFeatures let trueWeights, allData = makeAllData(numFeatures, numRows, seed) printfn "Data generation weights:" trueWeights |> Array.iter (printf "%.2f ") printfn "" printfn "Creating train (80%%) and test (20%%) matrices" let trainData, testData = makeTrainTest(allData, 0) printfn "Done" let maxEpochs = 1000 let alpha = 0.01 let classifier = Train (trainData,numFeatures,maxEpochs,alpha,0) let accuracy (examples:Example[]) = examples |> Array.averageBy (fun (feat,value) -> if classifier feat = value then 1. else 0.) accuracy trainData |> printfn "Prediction accuracy on train data: %.4f" accuracy testData |> printfn "Prediction accuracy on test data: %.4f"

We used a small trick to compute the accuracy – we mark every correct call as a one, every incorrect one as a zero, which, when we compute the average, gives us directly the proportion of cases that were called correctly. On my machine, I get the following output:

> Prediction accuracy on train data: 0.9988 Prediction accuracy on test data: 0.9980

Looks good enough to me, the implementation seems to be working. The whole code presented here is available as a gist here. I’ll leave it at that for now (I might revisit it later, and try to make this work with DiffSharp at some point, if anyone is interested) – feel free to ask questions, or drop me a comment on Twitter!

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